Controllability of Fractional Neutral Stochastic Integro-Differential Systems with Infinite Delay

This paper is concerned with the controllability of a class of fractional neutral stochastic integro-differential systems with infinite delay in an abstract space. By employing fractional calculus and Sadovskii's fixed point principle without assuming severe compactness condition on the semigroup, a set of sufficient conditions are derived for achieving the controllability result.


Introduction
It is well known that the fractional calculus is a classical mathematical notion and is a generalization of ordinary differentiation and integration to arbitrary (noninteger) order.Nowadays, studying fractional-order calculus has become an active research field [1][2][3][4][5][6][7].Much effort has been devoted to apply the fractional calculus to networks control.For example, Chen et al. [8], Delshad et al. [9], and Wang and Zhang [10] studied the synchronization for fractional-order complex dynamical networks; Zhang et al. [11] investigated a fractional order three-dimensional Hopfield neural network and pointed out that chaotic behaviors can emerge in a fractional network; Kaslik and Sivasundaram [12] discussed the local stability for fractional-order neural networks of Hopfield type by applying the linear stability theory of fractional-order system.
One of the emerging branches of this study is the theory of fractional evolution equations, say, evolution equations, where the integer derivative with respect to time is replaced by a derivative of fractional order.The increasing interest in this class of equations is motivated both by their application to problems from fluid dynamic traffic model, viscoelasticity, heat conduction in materials with memory, electrodynamics with memory, and also because they can be employed to approach nonlinear conservation laws (see [13] and references therein).In addition, neutral stochastic differential equations with infinite delay have become important in recent years as mathematical models of phenomena in both science and engineering, for instance, in the theory development in Gurtin and Pipkin [14] and Nunziato [15] for the description of heat conduction in materials with fading memory.It should be pointed out that the deterministic models often fluctuate due to noise, which is random or at least appears to be so.Therefore, we must move from deterministic problems to stochastic ones.We mention here the recent papers [16,17] concerning the existence of mild solutions of fractional stochastic systems.
As one of the fundamental concepts in mathematical control theory, controllability plays an important role both in deterministic and stochastic control problems such as stabilization of unstable systems by feedback control.Roughly speaking, controllability generally means that it is possible to steer a dynamical control system from an arbitrary initial state to an arbitrary final state using the set of admissible controls.Controllability problems for different nonlinear stochastic systems in infinite dimensional spaces have been extensively studied in many papers; see [18][19][20][21][22] and references therein.We would also like to mention that the controllability for stochastic systems with infinite delay has been investigated by Balasubramaniam et al. [23,24] and Ren et al. [25] using some abstract spaces.Nevertheless, to the best of our knowledge, it seems that little is known about the controllability of fractional neutral stochastic differential equations with infinite delay, and the aim of this paper is to close this gap.
In this paper, we are interested in the controllability of a class of fractional neutral stochastic integro-differential systems with infinite delay of the followin form: Here,  := [0, ],  > 0.   = {( + ),  ∈ (−∞, 0]} belong to the phase space  ℎ , which will be defined in Section 2. The initial data  = {(),  ∈ (−∞, 0]} is an F 0 -measurable,  ℎ -valued random variable independent of  with finite second moments, and  :  ×  ℎ → H,  :  ×  × H → L 0 2 (K, H) are appropriate mappings specified later (here, L 0 2 (K, H) denotes the space of all -Hilbert-Schmidt operators from K into H, which is going to be defined later).
The structure of this paper is as follows.In Section 2, we briefly present some basic notations and preliminaries.The controllability result of system (1) is investigated by means of Sadovskii's fixed point theorem and operator theory in Section 3. Conclusion is given in Section 4.

Preliminaries
For more details in this section, we refer the reader to Pazy [26], Da Prato and Zabczyk [27], and Samko et al. [28].Throughout this paper, (H, | ⋅ | H ) and (K, ‖ ⋅ ‖ K ) denote two real separable Hilbert spaces.We denote by L(K, H) the set of all linear bounded operators from K into H, equipped with the usual operator norm ‖ ⋅ ‖.In this paper, we use the symbol ‖ ⋅ ‖ to denote norms of operators regardless of the spaces potentially involved when no confusion possibly arises.
Let (Ω, F, {F  } ≥0 , ) be a filtered complete probability space satisfying the usual condition, which means that the filtration is a right continuous increasing family and F 0 contains all -null sets. = (  ) ≥0 is a -Wiener process defined on (Ω, F, {F  } ≥0 , ) with covariance operator  such that Tr  < ∞.We assume that there exist a complete orthonormal system {  } ≥1 in K, a bounded sequence of nonnegative real numbers   such that   =     ,  = 1, 2, . .., and a sequence of independent Brownian motions {  } ≥1 such that ) be the space of all Hilbert-Schmidt operators from  1/2 K to H with the inner product ⟨, ⟩ Suppose that 0 ∈ (−), where (−) is the resolvent set of −, then the semigroup (⋅) is uniformly bounded.That is to say, ‖()‖ ≤ ,  ≥ 0, for some constant  > 0.Then, for  ∈ (0, 1], it is possible to define the fractional power operator   as a closed linear operator on its domain D(  ).Furthermore, the subspace D(  ) is dense in H, and the expression defines a norm on H  := D(  ).The following properties are well known.
(a) If 0 <  <  ≤ 1, then H  ⊂ H  and the embedding is compact whenever the resolvent operator of  is compact.
(b) For every  ∈ (0, 1], there exists a positive constant   such that Let now us recall some basic definitions and results of fractional calculus.Definition 2. The fractional integral of order  with the lower limit 0 for a function  is defined as provided the right-hand side is pointwise defined on [0, ∞), where Γ(⋅) is the gamma function.
Definition 3. The Caputo derivative of order  with the lower limit 0 for a function  can be written as If  is an abstract function with values in H, then the integrals that appear in the previous definitions are taken in Bochner's sense.
Assume that ℎ : (−∞, 0] → (0, +∞) with  = ∫ 0 −∞ ℎ() < +∞ is a continuous function.Recall that the abstract phase space C ℎ is defined by If C ℎ is endowed with the norm then At the end of this section, we recall the fixed point theorem of Sadovskii [30].
Lemma 4. Let Φ be a condensing operator on a Banach space H; that is, Φ is continuous, and take bounded sets into bounded sets, and (Φ()) ≤ () for every bounded set  of H with () > 0. If Φ() ⊂  for a convex, closed, and bounded set  of H, then Φ has a fixed point in H (where (⋅) denotes Kuratowski's measure of noncompactness.)

Main Results
In this section, we obtain controllability of system (1).We first present the definition of mild solutions.Definition 5.An H-valued stochastic process {(),  ∈ (−∞, ]} is said to be a mild solution of system (1) if (i) () is F  -adapted and measurable for each  ≥ 0; (ii) () is continuous on [0, ] almost surely and for each  ∈ [0, ), the function ( − ) −1   ( − )(,   ) is integrable such that the following stochastic integral equation is verified: where with   a probability density function defined on (0, ∞).Definition 6. System ( 1) is said to be controllable on the interval , if for every initial stochastic process  ∈ C ℎ defined on (−∞, 0], there exists a stochastic control  ∈  2 (, U), which is adapted to the filtration {F  } ≥0 such that the mild solution () of ( 1) satisfies () =  * , where  * and  are preassigned terminal state and time, respectively.
The following properties of   () and   () that appeared in Zhou and Jiao [7] are useful.
(A 5 ) Assume that the following relationship holds: where Denote by ((−∞, ], H) the space of all continuous Hvalued stochastic processes {(),  ∈ (−∞, ]}.Let Set ‖ ⋅‖  to be a seminorm defined by We have the following useful lemma that appeared in Liu et al. [29].
Lemma 8. Assume that  ∈ C  ; then, for all  ∈ ,   ∈ C ℎ .Moreover, where The main object of this paper is to explain and prove the following theorem.
In what follows, we will show that using the control   (⋅), the operator  has a fixed point, which is then a mild solution for system (1).
For  ∈ C ℎ , define where  φ+ is obtained by replacing   by φ +   in (24).Let For each  ∈ C 0  , we have Thus, (C 0  , ‖ ⋅ ‖  ) is a Banach space.For  > 0, set Consider the map Π : C 0  → C 0  defined by A similar argument as (26), we can show that Π is well defined on   for each  > 0. Note that the operator  with a fixed point is equivalent to show that the operator Π has fixed point.To this end, we decompose Π as Π = Π 1 + Π 2 , where the operators Π 1 and Π 2 are defined on   , respectively, by (34) Thus, Theorem 9 follows from the next theorem.
Proof.The proof is followed by the several steps.
Step 1.There exists a positive number  such that Π(  ) ⊂   .If it is not true, then for each positive number , there exists a function   (⋅) ∈   , but Π(  ) ∉   ; that is, |(Π  )()| in view of Lemma 7 and Hölder inequality, we have where  0 and  are defined in ( 18) and (19), respectively.Applying Burkhölder-Davis-Gundy's inequality and assumptions (A 2 ), we get On the other hand, in view of ( 24) and (A 3 ), we have thus, by the same procedure as (36)-( 38), it follows that where 0 and  are defined in (18) and (19), respectively.Combining these estimates (35) to (40) yields where Dividing both sides of (42) by  and taking  → ∞, we obtain that which is a contradiction by assumption (A 5 ).Thus, for some positive number , Π(  ) ⊂   .
Step 2. Π 1 is a contractive mapping.Let , V ∈   .From the assumptions on  and , it is easy to verify that the following inequality holds: Thus, by the assumptions, we have where we have used the fact that  0 = V 0 = 0. Hence, so, Π 1 is a contraction by (23).
Step 3. We show that the operator Π 2 is compact.Let  > 0 be such that Π 2 (  ) ⊂   .The proof will be divided into the following claims.
Therefore, there are relatively compact sets arbitrary close to the set {Π 2 (),  ∈   }; hence, the set {(Π 2 )(),  ∈   } is also precompact in   .Thus, from Arzelá-Ascoli theorem together with assumptions on  and , we conclude that Π 2 is a compact operator.Therefore, Π is a condensing map on   .This completes the proof.
Remark 11.In order to describe various real-world problems in physical and engineering sciences subject to abrupt changes at certain instants during the evolution process, impulsive differential equations have been used to model the system.The technique used here can be extended to establish the controllability of neutral fractional stochastic integro-differential systems with impulsive effect and infinite delay.The controllability result can be obtained by suitably introducing the impulsive effects defined in [19].

Conclusions
In this paper, we have studied the controllability of fractional neutral stochastic integro-differential systems with infinite delay in an abstract space.Through fractional calculus and Sadovskii's fixed point principle, we have investigated the sufficient conditions for the controllability of the system considered.